If the lines \frac{x-1}{-3}=\frac{y-2}{2\lambda}=\frac{z-3}{2}\: \: and\: \: \frac{x-1}{3\lambda}=\frac{y-1}{2}=\frac{z-6}{-5}  are perpendicular, find the value of \lambda. Hence find whether the lines are intersecting or not.

 

 

 

 

 
 
 
 
 

Answers (1)

The equation of the given lines are:

\frac{x-1}{-3}=\frac{y-2}{2\lambda}=\frac{z-3}{2}\: \: and\: \: \frac{x-1}{3\lambda}=\frac{y-1}{2}=\frac{z-6}{-5}

If these lines are perpendicular, then

a_1a_2+b_1b_2+c_1c_2=0

\Rightarrow -3 \times 3\lambda +2\lambda \times 2+2\times \left ( -5 \right )=0

\Rightarrow -9\lambda +4\lambda -10=0

\Rightarrow -5\lambda =10

\lambda =-2

Now the lines are

\frac{x-1}{-3}=\frac{y-2}{2\lambda}=\frac{z-3}{2}\: \: and\: \: \frac{x-1}{3\lambda}=\frac{y-1}{2}=\frac{z-6}{-5}The coordinates of any point on first-line are given by:

\frac{x-1}{-3}=\frac{y-2}{-4}=\frac{z-3}{2}=\alpha

or x-1=-3\alpha \Rightarrow x=-3\alpha +1

y-2=-4\alpha \Rightarrow y=-4\alpha +2

z-3=2\alpha \Rightarrow z=2\alpha +3

So, coordinates of any point on this line are (-3\alpha+1, 4\alpha+2, 2\alpha+3)

The coordinates of any point on the second line are given by:

\frac{x-1}{-6}=\frac{y-1}{2}=\frac{z-6}{-5}=\beta

or x-1=-6\beta \Rightarrow x=-6\beta +1

y-1=2\beta \Rightarrow y=2\beta+1 z-6=-5\beta \Rightarrow z=-5\beta+6

So, coordinates of any point on second line are \left ( -6\beta+1, 2\beta+1, -5\beta+6 \right )

If lines intersect then they have a common point. So far some values of \alpha \: \: and\: \: \beta we have

-3\alpha+1= -6 \beta +1\Rightarrow -3\alpha= -6 \beta

\Rightarrow \alpha= 2 \beta

and \Rightarrow -4\alpha+2= 2 \beta +1

\Rightarrow -4\alpha+1= 2 \beta

On solving, we have 

\alpha =\frac{1}{5}\: \: \: and\: \: \beta=\frac{1}{10 }

The values of \alpha \: and\: \beta do not satisfy the third equation. 

Hence, lines do not intersect with each other.

 

Latest Asked Questions

Most Viewed Questions

Related Chapters

Preparation Products

Knockout CUET (Physics, Chemistry and Mathematics)

Complete Study Material based on Class 12th Syllabus, 10000+ Question Bank, Unlimited Chapter-Wise and Subject-Wise Mock Tests, Study Improvement Plan.

₹ 7999/- ₹ 4999/-
Buy Now
Knockout CUET (Physics, Chemistry and Biology)

Complete Study Material based on Class 12th Syllabus, 10000+ Question Bank, Unlimited Chapter-Wise and Subject-Wise Mock Tests, Study Improvement Plan.

₹ 7999/- ₹ 4999/-
Buy Now
Knockout JEE Main (Six Month Subscription)

- AI Coach Study Modules, - Unlimited Mock Tests, - Expert Mentorship, - Study Improvement Plan.

₹ 9999/- ₹ 8499/-
Buy Now
Knockout JEE Main (Nine Month Subscription)

- AI Coach Study Modules, - Unlimited Mock Tests, - Expert Mentorship, - Study Improvement Plan.

₹ 13999/- ₹ 12499/-
Buy Now
Knockout NEET (Six Month Subscription)

- AI Coach Study Modules, - Unlimited Mock Tests, - Expert Mentorship, - Study Improvement Plan.

₹ 9999/- ₹ 8499/-
Buy Now
Boost your Preparation for JEE Main with our Foundation Course
 
Exams
Articles
Questions